Steady Transonic Flow Research Articles

Nonlinear solution adaptive structured grids via heat-conduction analogies

This paper discusses an adaptive grid approach, developed using Fortran 77, on quadrilateral meshes for the Euler and Navier-Stokes solvers. Solution adaptation is through two nonlinear heat-conduction analogies applied directly on a two-dimensional surface using the finite volume method. Clustering of the grid generated is controlled by the conductivity in the computational domain, which is related arbitrarily to the geometrical curvature and flow gradient. Three levels of “multigrid” approach are implemented to accelerate convergence as a grid refinement process. The grid quality is accessed by a histogram analysis of maximum angle and aspect ratio distributions within the computational domain. This work assumes that interpolation errors due to numerical approximation of fluxes across the surfaces of a control volume should become significant as the skew angle and aspect ratio increases. Detailed computational results and comparisons with measured data are presented for steady transonic flow over a NACA0012 airfoil, supersonic flow through a DFVLR rotor, and a 15° ramp.

REDUCED-ORDER MODELS OF UNSTEADY TRANSONIC VISCOUS FLOWS IN TURBOMACHINERY

The proper orthogonal decomposition (POD) technique is applied in the frequency domain to obtain a reduced-order model of the unsteady flow in a transonic turbomachinery cascade of oscillating blades. The flow is described by a inviscid—viscous model, i.e. a full potential equation outer flow model and an integral equation boundary layer model. The nonlinear transonic steady flow is computed first and then the unsteady flow is determined by a small perturbation linearization about the nonlinear steady solution. Solutions are determined for a full range of frequencies and validated. The full model results and the POD method are used to construct a reduced-order model in the frequency domain. A cascade of airfoils forming the Tenth Standard Configuration is investigated to show that the reduced-order model with only 15–75 degrees of freedom accurately predicts the unsteady response of the full system with approximately 15 000 degrees of freedom.

Transonic flow of moist air around a thin airfoil with non-equilibrium and homogeneous condensation

A new small-disturbance model for a steady transonic flow of moist air with non-equilibrium and homogeneous condensation around a thin airfoil is presented. The model explores the nonlinear interactions among the near-sonic speed of the flow, the small thickness ratio and angle of attack of the airfoil, and the small amount of water vapour in the air. The condensation rate is calculated according to classical nucleation and droplet growth models. The asymptotic analysis gives the similarity parameters that govern the flow problem. Also, the flow field can be described by a non-homogeneous (extended) transonic small-disturbance (TSD) equation coupled with a set of four ordinary differential equations for the calculation of the condensate (or sublimate) mass fraction. An iterative numerical scheme which combines Murman & Cole's (1971) method for the solution of the TSD equation with Simpson's integration rule for the estimation of the condensate mass production is developed. The results show good agreement with available numerical simulations using the inviscid fluid flow equations. The model is used to study the effects of humidity and of energy supply from condensation on the aerodynamic performance of airfoils.

Computation of unsteady transonic flow over a fighter wing using a zonal Navier-Stokes/full-potential method

An improved hybrid method for computing unsteady compressible viscous flows is presented. This method divides the computational domain into two zones. In the inner zone, the Navier-Stokes equations are solved using a diagonal form of an alternating-direction implicit (ADI) approximate factorisation procedure. In the outer zone, the unsteady full-potential equation (FPE) is solved. The two zones are tightly coupled so that steady and unsteady flows may be efficiently solved. Characteristic-based viscous/inviscid interface boundary conditions are employed to avoid spurious reflections at that interface. The resulting CPU times are about 60% of the full Navier-Stokes CPU times for unsteady flows in non-vector processing machines. Applications of the method are presented for a F-5 wing in steady and unsteady transonic flows

Transonic shock oscillations on NACA0012 aerofoil

The mechanism of the origin of shock oscillations on NACA0012 aerofoils is investigated using a moving grid thin layer Navier Stokes code. The method used to understand the mechanism is to initiate the shock oscillations on an aerofoil by moving the aerofoil from a regime of steady transonic flow into a regime of periodic flow by a change in airflow incidence. The results indicate that the shock induced bubble plays a leading role in the origin of shock oscillations and the trailing edge has an affect on its amplitude.

A Staggered-Grid Multidomain Spectral Method for the Compressible Navier–Stokes Equations

We present a flexible, non-conforming staggered-grid Chebyshev spectral multidomain method for the solution of the compressible Navier–Stokes equations. In this method, subdomain corners are not included in the approximation, thereby simplifying the subdomain connectivity. To allow for local refinement of the polynomial order or subdomain subdivision, non-conforming subdomains are treated by a mortar method. Spectral accuracy is shown in one- and two-dimensional linear and non-linear problems. Application is made to four compressible flow problems: the Couette flow, a steady boundary layer over a flat plate, steady transonic flow in a nozzle, and unsteady flow over a cylinder at a Reynolds number of 75.

Longwave approximation model for gas shear flow in a channel of varying area

An approximate system of equations that describe unsteady flow of an inviscid non-heat-conducting gas in a narrow channel of varying area is derived. Generalized characteristics and hyperbolicity conditions are obtained for this system of equations. In connection with characteristics theory, the average Mach number and the flow criticality condition are introduced. Exact solutions that describe steady transonic channel flows are investigated.

STEADY TWO-DIMENSIONAL TRANSONIC ANALYSIS USING A BOUNDARY INTEGRAL EQUATION METHOD

A boundary integral equation method for the simulation of two-dimensional steady transonic potential flows is presented. The method is based on a conservative differential full-potential formulation. The steady two-dimensional formulation is obtained as the limiting case of the unsteady three-dimensional one, as the iterative method used to obtain steady-state results is a pseudo-time-accurate three-dimensional technique. In the present formulation the full-potential equation appears in the form of a nonlinear wave equation for the velocity potential. All the nonlinear terms, which are expressed in conservative form, are moved to the right-hand side, and treated formally as nonhomogeneous terms. The paper includes a historical review on the development of integral formulations for transonic analysis. Numerical results are obtained for steady two-dimensional transonic flows. Comparisons with existing finite-difference, finite-element, and finite-volume results shows a good agreement. The convergence analysis for an increasing number of grid elements reveals a limit behaviour in good agreement with other numerical methods in both subcritical and supercritical problems. Finally, we present numerical results to demonstrate a remarkable feature of the formulation, that is that the transonic numerical results are quite insensitive to the geometry of the field volume elements; this makes the present formulation particularly appealing for optimal design applications.

Global Structures of Advection-Dominated Two-Temperature Accretion Disks

Models of steady transonic global flows of optically thin, advection-dominated, two-temperature accretion onto black holes have been constructed. The physical quantities integrated over the vertical direction are treated with the Shakura-Sunyaev type α-viscosity. Bremsstrahlung and Compton cooling are considered, but synchrotron-Compton cooling is not taken into account. It has been found that the electron temperature in the inner part of the disk is fairly high in our model, because the surface density is low and the cooling there is rather inefficient. We have obtained Ṁcrit above which no steady solution is allowed, and have found that it is necessary to consider the relativistic cooling effect in the state that the electron temperature is higher.

Implicit time-stepping methods for the Navier-Stokes equations

A novel two-factor approximate factorization for three-dimensional flows is described. The method uses a preconditioned conjugate gradient solver for one of the factors, and this solution is based on a method developed for two-dimensional flows. This two-dimensional method is described, and its performance is summarized for steady and unsteady flows. The two-factor method is then given. Larger time steps can be used with this method than is possible with the more conventional three-factor method. Test results for the transonic steady flow over the ONERA M6 wing are given.

Tensor Universal Serendipity Elements and unsteady Taylor-Galerkin finite element method

A new kind of Universal Serendipity Element (USE)— the Tensor Universal Serendipity Element (TUSE) is constructed by using both tensor force finite elements and the basic idea of USE. The formulation of shape functions and their derivatives for TUSE is presented. TUSE can be used to study steady and unsteady transonic flow fields when combined with Taylor-Galerkin Finite Element Methods, the NND scheme in FDM, and four-stage Runge-Kutta methods. As numerical examples the transonic flow in cascades and one kind of complex unsteady transonic axisymmetric flow in engineering are studied. It is shown that the algorithm presented in this paper is efficient and robust.

Transonic Euler computation in streamfunction co‐ordinates

AbstractA new approach has been developed to calculate two‐dimensional steady transonic flows past aerofoils using the Euler equations in streamfunction co‐ordinates. Most existing transonic computation codes require the use of a grid generator to determine a suitable distribution of grid points. Although simple in concept, the grid generation may take a considerable proportion of the CPU time and storage requirements. However, this grid generation step can be avoided by introducing the von Mises transformation, which produces a formulation in streamwise and natural body‐fitting co‐ordinates. In this work a set of Euler equivalent equations in streamfunction co‐ordinates is formulated, consisting of three equations with three unknowns; one is a geometric variable, the streamline ordinate y, and the other two are physical quantities, the density ρ and the vorticity ω. To solve these equations, typedependent differencing, development of a shock point operator, marching from a non‐characteristic boundary and successive line overrelaxation are applied. Particular attention has been paid to the supercritical case where a careful treatment of the shock is essential. It is shown that the shock point operator is crucial to accurately capture shock waves. The computed results show excellent agreement with existing experimental data and other computations.

On steady transonic flow by compensated compactness

On steady transonic flow by compensated compactness

Numerical solution of 2D steady and unsteady transonic flows

Abstract The paper deals with modelling of transonic flows past a profile or through a cascade. The system of governing equations is the system of the Euler or Navier-Stokes equations. Numerical solution is based on finite volume form and MacCormack or multistep Runge-Kutta difference scheme (cell centered form) or Ron-Ho-Ni difference scheme (cell vertex form). Several numerical solutions of the steady transonic flows modelling inviscid flows through a 2D cascade or viscous flows past NACA 0012 are presented. The results are compared to experimental results or other numerical results. Next two types of modifications of the governing system of 2D Euler equations and its numerical solution are described. In this way certain three-dimensional phenomena are modelled.

Analysis of Steady and Unsteady Turbine Cascade Flows by a Locally Implicit Hybrid Algorithm

For the two-dimensional steady and unsteady turbine cascade flows, the Euler/Navier–Stokes equations with Baldwin-Lomax turbulence model are solved in the Cartesian coordinate system. A locally implicit hybrid algorithm on mixed meshes is employed, where the convection-dominated part in the flow field is studied by a TVD scheme to obtain high-resolution results on the triangular elements, and the second- and fourth-order dissipative model is introduced on the O-type quadrilateral grid in the viscous-dominated region to minimize the numerical dissipation. When the steady subsonic and transonic turbulent flows are investigated, the distributions of isentropic Mach number on the blade surface, exit flow angle, and loss coefficient are obtained. Comparing the present results with the experimental data, the accuracy and reliability of the current approach are confirmed. By giving a moving wake-type total pressure profile at the inlet plane in the rotor-relative frame of reference, the unsteady transonic inviscid and turbulent flows calculations are performed to study the interaction of the upstream wake with a moving blade row. The Mach number contours, perturbation component of the unsteady velocity vectors, shear stress, and pressure distributions on the blade surface are presented. The physical phenomena, which include periodic flow separation on the suction side, bowing, chopping and distortion of incoming wake, negative jet, convection of the vortices and wake segments, and vortex shedding at the trailing edge, are observed. It is concluded that the unsteady aerodynamic behavior is strongly dependent on the wake/shock/boundary layer interactions.

Euler study on porous transonic airfoils with a view toward multipoint design

Euler solutions for steady transonic flow (Free-stream Mach 0.63-0.8, alpha = 0-2 deg) over NACA 0012 and supercritical airfoils with solid as well as porous surfaces suggest porosity as a means to realize multipoint design for transonic airfoils. The porous surfaces extend over at least 90 percent of the chord. The porosity distribution is described by a modified sine wave with several amplitudes. Either connected or separated cavities are assumed to lie underneath the upper and lower surfaces. Applied to an NACA 0012 airfoil, porosity generally increases lift, in some instances by up to 65 percent. Porous NACA 0012 airfoils in supercritical flow yield reductions of an order of magnitude in wave drag at constant lift, compared to their solid counterpart. Making the surface of a supercritical airfoil permeable also leads to sizeable reductions in wave drag at constant lift for overspeed conditions. The discussion of the computed results addresses issues such as grid sensitivity and checks for systematic errors.

Weakly disturbed steady transonic flows of a vibrationally relaxing gas

The formation of shock waves in transonic flows is a problem which is far from being resolved. Conventional gas dynamics contains numerous examples of the construction of transonic flows free of discontinuities and flows in which discontinuities occur in the transition through the speed of sound (see [I], for example). Experiments have confirmed the existence of both continuous [2] transonic flows and transonic flows containing discontinuities [3]. The exceptional nature of nonshock flow in a local supersonic region was shown in [4], which suggested to one author that continuous stagnation of a transonic flow is not possible [5]. In the opinion Kuo and Sirs [6], such a flow is either unstable or contains shock waves or both. One possible approach to solving this problem is the study of transonic flows for their stability in relation to steady and nonsteady disturbances that might develop in the flow due to irregularities on the walls bounding the flow or the arrival of weak nonsteady waves at the sonic line. Such an investigation is complex in the general case, with approximate methods usually being used to obtain a solution. The stability of a transonic flow relative to small nonsteady disturbances in a small neighborhood of the sonic line was examined in [7, 8] on the basis of a quasi-unidimensional approximation. An attempt to perform an analysis in general form was made in [9], where results agreeing with the data in [7] were obtained after certain simplifying assumptions were made. Kuz'min [i0] analyzed the stability of a transonic flow on the basis of linearization of the Linn-Reissner-Tsien equation. It was found that transonic flow is stable in relation to a small change in the shape of the walls of the nozzle and the conditions at the inlet if the acceleration of the specified flow is everywhere positive. The examples of steady continuous transonic flows that have been constructed exist only near certain types of solid boundaries, thus giving rise to the view that such flows are the exception. A study of the behavior of steady perturbations of such flows due to irregularities in the walls bounding the flow will make it possible to study the development of the features of the flow. Aspects of the formation of shock waves and the conditions for nonshock flow in transonic flows were examined in [II]. It was shown that shock waves form if the slope of the walls of the channel near the mouth of the nozzle is too gentle. Discontinuities may form at a point on the sonic line or downstream, in the supersonic region. Numerical calculations of plane transonic flow with a local supersonic zone have shown that a shock wave is formed inside the region of supersonic flow [12]. Below, we examine the behavior of steady and nonsteady perturbations of steady transonic flows of a vibrationally relaxing gas. Use of the methods in [8] makes it possible to reduce the problem of determining the stability of a transonic flow against small nonsteady distortions to the study of a certain nonlinear partial differential equation. Analysis of the solutions of this equation for specific states of the medium shows that flow with a transition from the supersoni c regime to the subsonic regime is stable relative to small distortions if the gas is in the equilibrium state or if it is nonequilibrium with unexcited vibrational degrees of freedom. In this case, transonic flow will also be stable in the course of the transition from the subsonic to the supersonic regime. In our study of the behavior of steady distortions, we examined the supersonic region of transonic flow. We derive the equations of the transonic approximation as in traditional

Transonic flows of Bethe—Zel'dovich—Thompson fluids

Bethe–Zel'dovich–Thompson fluids are ordinary, single-phase fluids in which the fundamental derivative of gasdynamics is negative over a finite range of temperatures and pressures. We examine the steady transonic flow of these fluids over two-dimensional thin wings and turbine blades. The free-stream state is taken to be in the neighbourhood of one of the zeros of the fundamental derivative. It is shown that a modified form of the classical transonic small-disturbance equation governs such flows. Critical Mach number estimates are provided which take into account the non-monotone variation of the Mach number with pressure predicted in previous investigations. Critical Mach numbers well over 0.95 are predicted for conventional airfoil sections. Numerical solutions reveal substantial reductions in the strength of the compression shocks occurring in supercritical flows. Further new results include the prediction of expansion and compression shocks in the same flow and compression bow shocks in flows with subsonic free streams.

A comparison of adaptive‐grid redistribution and embedding for steady transonic flows

AbstractIn recent years, numerical analysts have increasingly turned to adaptive‐grid techniques to efficiently obtain highly‐accurate solutions to partial‐differential equations. This paper presents a comparison of two distinct grid‐adaptation approaches, grid‐point redistribution and grid‐embedding, applied to two‐dimensional, steady, inviscid, shocked flows. Grid redistribution is accomplished through a control function approach applied to the elliptic‐grid generator while the embedding approach consists of a fixed global grid and an automatically chosen number of irregularly shaped, embedded regions. For both techniques, the Euler flow equations are integrated to steady state using Ni's Lax‐Wendroff‐type integrator to predict the flow fields about a variety of isolated airfoil configurations. The application of a common integration scheme as well as a common adaptive control function leads to a direct comparison of the effectiveness of the two adaptation approaches for the types of problems tested.

Transonic flow of a fluid with positive and negative nonlinearity through a nozzle

The one-dimensional transonic flow of an inviscid fluid, which at large values of the specific heats exhibits both positive (Γ̄>0) and negative (Γ̄<0) nonlinearity regions {Γ̄=(1/ρ)[∂(ρa)/∂ρ]s} and which remains in a single phase, is studied. By assuming that Γ̄ changes its sign in the small neighborhood of the throat of the nozzle where transonic flow exists and introducing a new scaling of the independent variables, an approximate first-order partial differential equation (PDE) with a nonconvex flux function is derived. It governs both the steady transonic flows and the upstream moving waves near sonic point. The existence of continuous and discontinuous steady transonic flows when the throat area is either a maximum or a minimum is shown. The existence of standing sonic discontinuities and rarefaction shocks in the transonic flow are noted for the first time. Unlike in the classical gas flows, there are two sonic points and continuous transonic flows are possible only through one of them. The numerical evolution of those transonic waves that have both positive and negative nonlinearity in the same pulse is studied and some comments are made on the local stability of the particular steady flows.